A footpath of uniform width runs round the inside of a rectangular field 32 m long and 24 m wide. If the path occupies 208 $m^{2}$ , find the width of the footpath.

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Solution

Let w be the width of the footpath.

$A r e a o f t h e p a t h = A r e a o f o u t e r r e c t a n g l e - A r e a o f i n n e r r e c t a n g l e$
∴ $208 = (32) (24) - (32 - 2 w) (24 - 2 w)$
$208 = 768 - 768 + 64 w + 48 w - 4 w 2$
$4 w^{2} - 112 w + 208 = 0$
$w^{2} - 28 w + 52 = 0$
$w^{2} - 26 w - 2 w + 52 = 0$
$w (w - 26) - 2 (w - 26) = 0$
$(w - 26) (w - 2) = 0$
w = 26, 2
If w = 26, then breadth of inner rectangle = (24 – 52) m = -28 m, which is not possible.
Hence, the width of the footpath is 2 m

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A footpath of uniform width runs round the inside of a rectangular field 32 m long and 24 m wide. If the path occupies 208 m2, find the width of the footpath.

A footpath of uniform width runs round the inside of a rectangular field 32 m long and 24 m wide. If the path occupies 208 $m^{2}$ , find the width of the footpath.